Low-rank multi-parametric covariance identification

• Main Authors: Musolas, Antoni (Author), Massart, Estelle (Author), Hendrickx, Julien M. (Author), Absil, P.-A (Author), Marzouk, Youssef (Author)
• Format: Article
• Language: English
• Published: Springer Netherlands, 2022-02-02T15:48:09Z.
• Subjects: Article
• Online Access: Get fulltext

 Abstract We propose a differential geometric approach for building families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of "anchor" matrices for interpolation, for instance over a grid of relevant conditions describing the underlying stochastic process. The interpolation is computationally tractable in high dimensions, as it only involves manipulations of low-rank matrix factors. We also consider the problem of covariance identification, i.e., selecting the most representative member of the covariance family given a data set. In this setting, standard procedures such as maximum likelihood estimation are nontrivial because the covariance family is rank-deficient; we resolve this issue by casting the identification problem as distance minimization. We demonstrate the utility of these differential geometric families for interpolation and identification in a practical application: wind field covariance approximation for unmanned aerial vehicle navigation.